Let $K$ be a local field, $K^{sep}$ its separable closure, $G = Gal(K^{sep}/ K)$ the Galois group and $C := \overline{K^{sep}}$ the completion with respect to the induced valuation.

In his paper on $p$-divisible groups, Tate proves that if $K$ is a $p$-adic field, then the continuous cohomology groups $H^{i}(G, C)$ are one-dimensional over $K$ when $i = 0, 1$ and vanish otherwise.

Are these continuous cohomology groups known when $K \simeq \mathbb{F}_{q}((t))$ is non-Archimedean of equal characteristic?